Apparatus and method for grinding irregular surfaces of revolution

ABSTRACT

A workhead, for the grinding of non-circular workpieces, equipped with a pair of compensating cams to shift the workpiece axis relative to the axis of the grinding wheel. The &#39;&#39;&#39;&#39;X&#39;&#39;&#39;&#39; cam shifts the workpiece axis horizontally; the &#39;&#39;&#39;&#39;Y&#39;&#39;&#39;&#39; cam shifts it vertically. The rotation of work and cams is synchronized so that the line jointing the point of grind, the center of the grinding wheel and the instant center of curvature of the desired surface is coincident with the feed and compensation line of the grinder at all times.

United. States Patent Cann [451 Sept. 26, 1972 [54] APPARATUS AND METHOD FOR GRINDING IRREGULAR SURFACES OF REVOLUTION [72] Inventor: Roald Cann, Weathersfield, Vt.

[73] Assignee: Bryant Grinder Springfield, Vt.

March 24, 1970 Corporation,

[22] Filed:

v[21] App]. No.: 22,307

[52] U.S.Cl. ..51/94 R,51/95 WH,51/101R [51] Int. Cl. ..B24b 17/02, B24b 5/16, B24b 5/36 [58] Field of Search ..51/93, 94 R, 95 R-97, 51/50 PC, 237 R, 290, 281 R, 281 P, 281 C,

101 R, 101 LG, 105 R, 105 EC, 95 WI-l [56] References Cited UNITED STATES PATENTS 2,415,062 l/1947 Green ..51/101 2,452,989 11/1948 Brown ..51/101 2,487,201 1l/l949 Van Buren ..51/101 X 2,592,875 4/1952 Durland ..93/ 2,606,403 8/1952 Musyl ..51/101 X FOREIGN PATENTS OR APPLICATIONS 662,769 12/ 1 951 Great Britain ..51/101 Primary Examiner-Donald G. Kelly Attorney-James H. Bower [57] ABSTRACT A workhead, for the grinding of non-circular workpieces, equipped with a pair of compensating cams to shift the workpiece axis relative to the axis of the grinding wheel. The X cam shifts the workpiece axis horizontally; the Y cam shifts it vertically. The rota tion of work and cams is synchronized so that the line jointing the point of grind, the center of the grinding wheel and the instant center of curvature of the desired surface is coincident with the feed and compensation line of the grinder at all times.

9 Claims, 13 Drawing Figures llll ll Hill ,1

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INVENTOR ROAAD CA/V/V ATTORNEYS APPARATUS AND METHOD FOR GRINDING IRREGULAR SURFACES OF REVOLUTION BACKGROUND OF INVENTION This invention concerns an apparatus for grinding workpieces of revolution having variable radii, such as cams, but relates more particularly to the workhead for revolving and positioning the workpiece relative to the grinding wheel.

It is well known in the art to grind cams and the like by rotating the workpiece while translating its center of revolution. in order to keep the work profile in constant contact with the grinding wheel. One such arrangement successfully used is disclosed in U.S. Pat. No. 2,836,936; issued June 3, 1958 to John Lovely, another arrangement is shown in U.S. Pat. No. 2,592,875, issued Apr. 15, 1959 to Philip'Durland.

These mechanisms are very satisfactory for grinding non-circular shapes in which the center of curvature always stays on the same side of the surface being ground, such as elliptical figures. They are not well adapted to reverse curves in which the center of curvature crosses over to the opposite side, or where there are sudden changes in direction, or severe radial changes. Any reverse curvature causes extreme variation in relative work surface to wheel speed along the grinding surface. Concave portions of the workpiece contour cause increased linear speed of the grinding wheel/workpiece contact point along the ground surface (LSCP), resulting in less stock removal; convex portions cause decreased LSCP and a tendency toward burning of the surface. It can readily be seen that the linear speed must be controlled if accurate parts are to be ground.

Therefore one object of the invention is to hold the linear speed of the contact point (LSCP) around the workpiece contour to a nearly constant value.

Another important consideration in repetition of the precise shape to be ground is the grinding wheel diameter. If the wheel diameter could be kept constant, it would be possible to design positioning cams to precisely locate the center of the wheel at all times so as to produce identical workpieces. Any variation of wheel diameter, however, defeats this purpose by producing a slightly different shape. Since the contact point between the wheel and workpiece is changing from workpiece to workpiece, due to wheel wear, it is usual to design the cams for an average wheel diameter, thus keeping the ground shape within tolerance.

Certain workpieces have extremely small tolerances. It is therefore another object of the invention to eliminate the effect of wheel wear on the shape of ground surface so as to fully utilize the wheel diameter range.

I SUMMARY OF THE INVENTION The solution to the above stated problems lies in the precise location of the instant center of workpiece revolution. Proper variation of the location of the instant center of revolution as the workpiece rotates, is accomplished through a unique combination of X and Y cams in constant contact with a sliding plate on which the workhead chuck is journalled for rotating the workpiece.

The X and Y cams are designed to keep the line on which are located the instant center of curvature,

the center of the grinding wheel, and the contact point between wheel and workpiece, always coincident with the feed and compensation line of the grinding machine.

As the motor rotates the cams and the workpiece, the center distance between the X" cam and X cam follower and between the Y" cam and the Y cam follower changes in accordance with the contours of the X and Y" cams, respectively. The X cam causes essentially translatory motion of the workpiece in the X direction. The Y cam causes both an approximately translatory motion of the workpiece in the Y direction plus a very important rotary or pivotal motion of the wo'rkspindle housing.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 is perspective view of an automatic grinding machine incorporating the invention;

FIG. 2 is enlarged view of the grinding area;

FIG. 3 is typical workpiece showing the geometrical centers of curvature and contact points between whee and workpiece;

FIG. 4 is the workpiece showing the change in ground shape between maximum and minimum diame- DESCRIPTION OF THE PREFERRED EMBODIMENT Referring first to FIG. 1, we see an automatic precision contour grinding machine 10, having a base 11, on which are mounted a wheelslide 12, adapted for reciprocation longitudinally as indicated by arrow 14. On the left end of the machine is carried the workhead 16, part of which is shown through the glass window 18 in the splash guard 20.

Also seen through the window 18 is the grinding wheel 22 and the grinding wheel dresser 24, in raised position.

FIG. 2 is a larger view of the grinding area with the splash guard 20 removed, and shows the workhead l6, and the grinding wheel 22, and the grinding wheel dresser 24. The grinding wheel22 is mounted on the outer end of the quill or wheel spindle 28, supported and rotated in bearings (not shown) inside the wheelhead 26.

Also seen in FIG. 2 is the chuck 30, having three fluid operated clamping fingers 32 holding workpiece 34 down on two locating dowel pins (not shown). The chuck 30 is rotatable, as indicated by arrow 31, and translatable radially as will be described.

FIG. 3 shows thelayout of a typical workpiece 34, identifying the centers of curvature and inflection points of the interior surface 36.

The typical ground interior surface 36 chosen for our illustration is a somewhat synthesized Wankel engine cylinder. The surface to be ground resembles a figure eight" in some respects, with radial arcs and corresponding centers as follows:

Arc from 2 to 4 with center of curvature at a Arc from 4 to 6 with center of curvature at d Arc from 6 to 8 with center of curvature at b Arc from 8 to 2 with center of curvature at c The largest grinding wheel 22 shown is of radius R centered at V, while the smallest grinding wheel 23 of radius r centered at W.

The surface 36 is symmetrical about a center at 0, which is also the axis of the chuck rotation. The lower lobe I and the upper lobe II are symmetrical in the particular shape chosen, and therefore the specification need be concerned with only one of these lobes.

FIG. 4 shows the surface 36 as generated by a machine having a X cam only; no Y cam correction used. That is, a machine using the concept of generat ing non-circular profiles or surfaces with only a single cam.

Assume that thegrinding starts at the upper lobe II of the workpiece 34, on the surface 36 at point 3. With the workpiece 34 rotating at a constant rate about the center in the direction shown by arrow 31, the X cam positions the wheel center at V a distance X from a base point or axis of chuck rotation, O, on the workpiece. With the locus 37 of the wheel center V shown for the upper lobe II, the wheel of diameter 2R then generates the surface 36 starting at point 3.

As the chuck rotates the workpiece 34 at a uniform angular rate, the X cam, which is timed to keep step with the workhead rotation, constantly applies a correction to the radial distance X.

' It should be noted that while point 3 and point 4 are close together linearly along the surface 36, in comparison with the linear distance from point 4 to point 5, the workpiece is rotated more than half the angular distance, or about 51. In other words, the workpiece is rotated approximately 51 from point 3 to point 4, but only 39 from point 4 to point 5. It is therefore evident that the linear speed of the contact point (LSCP) varies considerably along the surface to be ground; which creates an undesirable condition.

The second Quadrant, points 5-6-7, will be the same as the first Quadrantpoints 3-4-5; but in reverse order. That is, the linear speed will be faster from point 5 to point 6 than from point 6 to point 7.

A study of FIG. 4 will show that points 1, 3, 5, and 7 are all in direct radial lines extending through their wheel centers V V V or V, and through the workpiece center of rotation O. The other points in between are. not on such a straight line; namely, center 0, V.,, and point 4.

A second undesirable condition becomes evident when the wheel has worn smaller as; for example, when radius R becomes r. This requires a compensation in the amount of X an extension of the X correction. The wheel center then becomes W W W W W and so on.

With the compensation X always along the radial line from the center of rotation O, the locus of center W is 38, and the small wheel then generates the shape shown as the envelope 39. It is readily seen that this envelope is markedly different from surface 36. For production grinding, this condition is totally unacceptable.

With reference now to FIG. 5. The workpiece 34 still will have the surface to be generated by the grinding wheel of radius R. Starting at point 2, the grinding wheel center is successively at V V V etc. There is a noticeable difference, however, from F IG. 4. The radial line from the instant center of curvature, to the center of the grinding wheel, is now maintained coincident with the feed line; this was not so as shown in FIG. 4. This is possible by adding a Y cam correction to each successive point, as will be hereafter described. For the purposes intended in this application, X is defined as the correction distance along the feed line, and Y as the correction distance at right angles to the feed line.

Consider first, point 2. As in FIG. 4, the grinding wheel center is shown in FIG. 5 at V,. In FIG. 4, however, the feed line is coincident with the line V -O and the point 2 is ground by the side" of the wheel; i.e., a point not on the feed line.

In FIG. 5, the feed line for point 2 is a-2-c, which is coincident with the radial line 2V The X distance from V to the center 0, using the former method, was

called X Now, since by definition,X is the correction along the feed line, and Y is the correction at right angles to the feed line, the radial line V -O is replaced by the two coordinates X and Y For convenience the line designated V -O is Z (the same line was designated X in FIG. 4).

Using the same procedure for each point in succession, the grinding wheel center is located by the two coordinates relative to the workpiece center, 0. This is further illustrated at points 3 and 4. At point 3, the feed line is a-3O, and therefore the point V as before, is located at X and Y (Y here equals zero). Consequently, Z equals X for this point. At point 4, radial line 2., breaks down to X, and Y from which point around to point 6 the center of curvature remains in the same position with respect to the surface 36, and noted as a. The wheel centers are indicated as V V and V Point 6 is the same as point 2 but from it.

As in FIG. 4, when the grinding wheel wears down in diameter, the loss in radius must be compensated, and the amount of compensation will again be called X Since X is always along the feed line, it adds to X directly and does not affect Y. This retains the contact point of the worn grinding wheel at the same point as that of a larger grinding wheel. Thus the surface 36 is retained and is independent of wheel diameter.

The locus of the instant centers V of the large wheel, as it grinds each point, is shown as 41, and that of the centers, W, of the small wheel as 42. It is clearly seen that these two loci are parallel, and spaced exactly a distance, X apart.

As stated earlier, one important object of this inven" tion is to control the linear speed of the contact point (LSCP) along the surface of the workpiece. This is accomplished by precisely located the center 0 of the surface at all times so that equal incremental linear distances along the surface will pass the grinding point in approximately equal increments of time. Proper shape and design of the X and Y cams and selection of the dimension S, (the center distance between the work spindle O and the lower pivot shaft 61, seen in FIG. 12) will insure this.

To determine S, the following procedure is important: Calculate the LSCP at various points along the surface of the workpiece for each of several 8- values, positioning the workpiece incrementally by graphical methods or by digital computer, and determining the resulting locations of the other moving parts. -By plotting the curves LSCP vs Polar angle about the S value curve showing least variation can be readily apparent.

The dimension S can also be determined approximately as the average of the major and minor distances from the ground surface to the center 0, i.e., for FIG.

S E (0 O )/2 Where the distance S is defined as the center distance between the center axis of the workpiece O and the lower pivot shaft axis 61, 0 as the workpiece major radius, and O, as the workpiece minor radius. This formula will be satisfactory for most purposes.

Throughout most of the above noted specification the different points have been considered around a workpiece which has been held in one position. In reality, the opposite is true; the workpiece is rotating about a center 0, which is simultaneously moved up, down, left, and right relative to the wheel center V. These various positions are illustrated in FIGS. 6 to 11, which will now be explained.

FIG. 6 shows the grinding wheel contacting point 1 at lobe I of the workpiece 34. The wheel center is shown as W. Since this is one of the four points around the surface 36, (ll, 3, 5, and 7), where the feed line and line of compensation 43 are coincident with the line joining the contact point and the centers of curvature and rotation, no Y correction is necessary, only X. The relative positions are emphasized clearly by the small diagrams to the right indicating the relative positions of the center of curvature with respect to the workhead center 0 and wheel center W.

As the workpiece rotates through the next position, FIG. 7, the surface from point I to point 2 is ground. Hereafter the feed line 43 coincides with the line joining the contact point 2 and the center of curvature c, therefore a Y correction and an x vcorrection are applied to shift the work center 0. The path of the center 0, and thus the workpiece, as shown by the dotted arrow, isradiused about 0 in the small right hand diagram of FIG. 7.

It should be noted that there are four centers of curvature: two external and two internal. The two external centers of curvature are at a and b, and the two interior centers of curvature are at c and d.

As shown in FIG. 3, FIG. 4, and FIG. 5 points 2, 4, 6, and 8 are the points of inflection. The point of inflection is where there is a transference of centers ofcurvature. That is, as shown at point 2, in FIG. 7, the interior "center of curvature c is transferred to the external center of curvature 0. Regions having an external center of curvature are sometimes referred to as regions of reverse curvature. It is interesting to note that at the point of inflection, the excursion of the center 0 seems to reverse direction. This is shown In FIG. 8 and FIG. 9 by the dotted arrows (radiused about a) where the workpiece appears to reverse direction and shift backwards. Nevertheless the LSCP-linear speed of the contact point-along the surface of the workpiece is constant and controlled.

The X dimension is changing continuously from X shown in FIG. 7, through X, to X,, as shown in FIG. 8 and FIG. 9; simultaneously, the Y dimension is changing from Y through zero to Y.,.

FIG. 9 shows point 4 as a point of inflection, where the center of curvature transfers from a to d. This is similar to FIG. 7 where point 2, as a point of inflection, is the terminal point with the center of curvature at c and the initial point with the center of curvature at a. From FIG. 9 through FIG. 11, the center of curvature remains at d. The center of curvature transfers to b at point 6 and remains so until point 8, as shown in FIG. 1 1.

During this continuous series of excursions, while X and Y dimensions are continuously changing, it is to be noted that the workpiece is rotating at a constant rate about center 0 in the direction indicated by arrow 31. This compound motion might be likened to a wheel rolling along a bumpy road, constantly rotating, yet moving forward, and up and down as well.

A comparison of FIG. 6 to FIG. 10 and FIG. 7 to FIG. 11 will show their similarity. In the comparison of FIG. 6 to FIG. 10: X dimension equals X Y and Y dimensions are equal to zero and therefore not shown. The similarity of FIG. 7 to FIG. 11 shows that X dimension equals X Y dimension equals Y Thus, a symmetrical workpiece can be designed with translation cams X and Y for fa a revolution of the workpiece and there rotation synchronized with the workpiece at 2:1 ratio. This allows more accuracy by doubling the cam size.

Referring now to FIG. 12, a phantom schematic view of the workhead 16 is shown from the wheelhead side. The workpiece, centered at O, mounted in chuck 30 (as seen in FIG. 2), is not shown here but rotates with work spindle shaft 45. The work spindle rotates on bearings mounted in the work spindle housing 44, which is free to float up and down or laterally on two opposing sets of hydrostatic bearings 46, one such set of three being indicated.

As workspindle housing 44 floats, it must be restrained to constantly position the workpiece center 0 at the correct locations, synchronized with the angular rotation of the workpiece. This is accomplished by a pair of plate cams X cam 48 and Y cam 50 and work spindle support link 58. The cams are both mounted on cam shaft 52, rotating on bearings mounted in'workhead housing 54.

At the upper edge of the workhead housing 54 is fixedly located upper pivotshaft 56, which carries the work spindlesupport link 58, (a bell crank pivoted at its center). The work spindle work housing 44 is sup ported by means of lower pivot shaft 61, which also carries X follower 6.0, so that the work spindle center 0 can slide laterally as the work spindle support link 58 swings about pivot shaft 56, with each X cam radius change.

The X cam follower 60 is always retained against the X cam 48, biased downwards by hydraulic X cylinder 66 operating at the other end of the work spindle support link 58.

The work spindle housing 44 is adapted to pivot about shaft-61, under control of the Y cam follower 62, on Y follower shaft 63 as the radius of the Y cam 50 varies. The Y cam follower 62 is always retained against the Y cam 50 by hydraulic Y cylinder 68 biased against an extension of work spindle housing 44.

As the motor rotates the X and Y cams and the workpiece, the center distance between the X cam and the X cam follower changes in accordance with the contour of the X cam. Since the X cam follower is journalled at 61 to the work spindle housing 44, the work spindle housing 44 translates in the X direction as permitted by link 58 which is connected to the work spindle housing by lower pivot shaft 61 and to workhead housing 54 by upper pivot shaft 56. As the motor rotates the X and Y cams, and the workpiece relative to the work spindle housing 44, the center distance between the Y cam and the Y cam follower changes in accordance with the contour of the Y cam. Since the Y cam follower is journalled on the work spindle housing 44 by Y follower shaft 63, work spindle housing 44 moves as permitted by lower pivot shaft 61-. This motion, unlike the essentially translatory X direction motion, consists of both an approximately translatory motion of the workpiece center in the Y direction, and a very important rotary or pivotal motion of the work spindle housing 44. As the Y cam rotates continuously in one direction, Y follower 62 alternately is' pushed down by a so-called rise in the contour of Y cam 50 or travels up on a so-called fall in the contour, causing work spindle housing 44 to pivot alternately clockwise or counterclockwise about lower pivot shaft 61. The contour of Y cam 50 is designed to cause a maximum clockwise pivoting velocity of work spindle housing 44 which is greater than the constant counterclockwise rotational velocity of the workpiece with respect to work spindle housing 44 as imposed by the motor via worm 73 and worm wheel 72. Therefore, during times of maximum and near-maximum clockwise pivoting velocity of work spindle housing 44, the pivoting motion more than counteracts the steady counterclockwise rotation, and during those times the workpiece actually rotates in the clockwise direction. Thus, it is shown that the rotation of the workpiece periodically reverses even though the motor rotates continuously at constant speed and direction.

In the schematic shown FIG. 12, the X cam 48 is shown as a circle. Actually, the shape varies as the X dimension varies; a reducing radius produces a reducing X correction, and vice versa. Likewise, the reducing radius on the Y cam lowers the center 0, producing a minus Y and an increasing radius moves center to a plus Y correction. Needless to say, the X and Y cams must be designed as a unit for any given work shape.

Clearly, neither cam may contain regions of Switchback, also known as three-valued or hooked regions, since such regions cause the follower either to jam or to skip, depending on the direction of cam rotation. The pivoting of work spindle 44 about lower pivot shaft 61, which has already been shown to produce the necessary reversals in the direction of workpiece rotation, results in variations between the angular orientation of the workpiece 34 and the angular orientation of the Y cam 50. This assures that those points on the workpiece which require identical workpiece angular orientations, for example, the major axis oriented vertical, will be ground with the Y cam follower contacting the Y cam at non-identical angular orientations of the Y cam, thereby precluding the need for a Switchback in the Y cam contour.

It is imperative that the rotation of the work spindle carrying the workpiece must be synchronized with the rotation of the cam shaft if the correct shape is to be achieved. FIG. l3 discloses the mechanism by which the preferred embodiment is synchronized. This view is taken from the rear of the workhead mechanism with covers removed, directly behind that of FIG. 12, looking toward the wheelhead. The other end of the work spindle shaft 45 is shown centered at O, and of the cam shaft 52. On each shaft is mounted a worm wheel, both of the same pitch diameter and number of teeth. A double threaded worm 71 drives worm wheel keyed to cam shaft 52 by key 53, and a single threaded worm 73 drives worm sheel 72 keyed to the work spindle shaft 45 at O by key 77. Thus, the drive ratio is 2:], so that one revolution of the cam shaft controls only one half revolution of the workpiece. The second half of the ground surface is identical to the first and is controlled by a second revolution of the cams.

It should be understood that if the workpiece was not symmetrical as in the embodiment used, the cams would have to rotate at the same rate as the workpiece, and both worms would be single thread to keep a 1:1 ratio.

The power to operate the workhead is obtained from an electric motor, (not shown), and is transmitted'to the worm drive through a belt 74 and pulley 75, on the.

worm shaft 76, journalled in worm housing 78, which is fixed to the base 11. On the right end of the worm shaft 76 is attached a flexible spring disc coupling 80. On the outside of the coupling 80 is attached a ball spline 82 and the spline shaft 84 which slides in and out of the ball spline 82. The other end of spline shaft 84 is flexibly attached to worm 73 through constant velocity universal joint 86.

Worm 73 is mounted for rotation in bearing bracket 88. Since worm 73 must always mesh with worm wheel 72, and since worm wheel 72 is keyed at 77 to work spindle shaft 45, bearing bracket 88 is integral with floating bearing plate 89 and with work spindle housing 44 to assure that worm 73 makes the same horizontal and vertical excursions as does worm wheel 72. These excursions are permitted by opposed hydrostatic bearings 46, described earlier.

As these excursions take place, the spline shaft 84 (slidable in the ball spline 82), the universal joint 86, and the flexible spring disc coupling 80 accommodiate the horizontal and vertical movements. Though both worms and worm wheel 70 rotate at a constant rate,

driven by belt 74 and pulley 75, worm wheel 72 rotates at a varying rate as a result of the cyclic excursions of worm 73 about center 0. The variable rate of rotation of worm wheel 72, and therefore of workpiece 34, can be observed (see FIGS. 6 through 11) to include apparent reversals in the direction of rotation.

Having disclosed the specification of the preferred embodiment of my invention, the mechanism may be summarized as follows: The work holding chuck carrying a workpiece having a smoothly varying closed surface of non-circular form to be ground at a nearly uniform linear speed of the grinding wheel/workpiece contact point, is movable both laterally and vertically while continuously rotating, in order to apply the grinding wheel directly along the feed line, thus making the size and shape of ground surface portion entirely independent of wheel size.

It should be understood that I have described only a preferred embodiment of my cam grinder, other arrangements being possible within the spirit and scope of the invention.

I It is claimed:

1. An apparatus for grinding non-circular workpieces of revolution, comprising:

a. a workhead having a holding means adapted to grip a workpiece with a major axis and a minor axis and rotate said workpiece about a center axis, said holding means mounted on a work spindle journalled in a work spindle housing (44);

. an X cam and a Y cam journalled for rotation on a shaft parallel to said work spindle, and rotating in timed relation therewith;

c. a grinding wheel, said grinding wheel adapted to contact said workpiece at a point spaced from said center axis;

. a pair of X and Y cam followers attached to said work spindle housing and maintained in constant contact with said X and Y cams, respectively;

1. said X cam follower adapted to move said workpiece in a direction toward or away from said grinding wheel, said direction defined as the X direction;

2. said Y cam follower adapted to impart pivoting motion to said workhead about a point on said workhead, said point being located from said center axis of said workpiece a distance not less than the minor radius of said workpiece, nor greater than the major radius of said workpiece, in the direction of the contact point between said workpiece and said grinding wheel 2. A workhead for rotating about a center axis and shifting the axis of rotation of a non-circular workpiece having both a major and a minor axis, relative to a grinding wheel axis of rotation, comprising:

a. a grinding wheel;

b. a workspindle housing adapted to move in an X direction;

c. said work spindle housing also adapted to be pivotally movable;

d. X and Y cams rotated in timed relation'to the rotation of said non-circular workpiece to produce work spindle housing movement; v

e. the axis of said pivotal motion being located from the center axis of said workpiece a distance not less than the workpiece minor radius nor greater than the workpiece major radius, in the direction of the contact point between said workpiece and said grinding wheel.

3. The invention defined in claim 1 wherein:

a. the distance of said point from said workpiece center axis is structurally defined by the formula:

where the distance S is defined as the distance between the center axis of said workspindle and the lower pivot shaft axis (61), 0, as the workpiece major radius, and 0 as the workpiece minor radius.

4. The invention defined in claim 2, wherein: a. The distance of said centerline from said workpiece center axis is structurally defined by the formula:

where the distance S is defined as the distance between the center of axis of said workspindle (0) and the lower pivot shaft axis (61), 0, as the workpiece major radius, and 0 as the workpiece minor radius.

5. A method of grinding a non-circular workpiece, having regions of reverse curvature with both a major axis and a minor axis, by rotating said workpiece about a center axis, comprising the steps of:

a. moving two cams in rotation about a fixed axis;

b. moving one of said rotatable cams thereby effecting movement of said workpiece in an X direction;

c. moving the other one of said rotatable cams thereby effecting movement of said workpiece in a transverse direction;

1. said transverse movement of said workpiece being a pivoting motion about a point to locate the instant center of curvature of said workpiece at all times on the feed line, said feed line is defined as the line through the grinding wheel center axis in thedirection of compensation;

2. Locatingsaid point from said workpiece center axis by a distance which is derived by the formula:

where the distance S is defined as the distance between the center axis of said workspindle (0) and the lower pivot shaft axis (61), 0, as the workpiece major radius, and 0 as the workpiece minor radius.

6. A method of grinding a non-circular workpiece, having regions of reverse curvature with both a major axis and a minor axis, by rotating said workpiece about a center axis, comprising the steps of:

a. moving two cams in rotation about a fixed axis;

b. moving one of said rotatable cams thereby effecting movement of said workpiece in an X direction;

c. moving the other one of said rotatable cams thereby effecting movement of said workpiece in a transverse direction;

1. said transverse movement of said workpiece being a pivoting motion about a point to locate the instant center of curvature of said workpiece at all times on the feed line, said feed line is defined as the line through the grinding wheel center axis in the direction of compensation.

7. A method of grinding a non-circular workpiece having regions of reverse curvature, with both a major axis and a minor axis, by rotating said workpiece about a center axis, comprising the steps of:

a. moving two cams in rotation about a fixed axis;

b. moving one of said rotatable cams thereby effecting movement of said workpiece in an X direction;

c. moving the other one of said rotatable cams thereby effecting movement of said workpiece in a transverse direction;

1. said transverse movement of said workpiece being a pivoting motion about a point to locate the instant center of curvature of said workpiece at all times on the feed line, said feed line is defined as the line through the grinding wheel center axis in the direction of compensation;

2. locating said point (61) by that distance from the center axis of said workpiece toward the where the distance S is defined as the distance between the center axis of said workspindle (0) and the lower pivot shaft axis (61 0, as the workpiece major radius, and 0 as the workpiece minor radius.

9. A method of grinding a non-circular workpiece, having regions of reverse curvature with both a major axis and a minor axis, by rotating said workpiece about a center axis, comprising the steps of;

a. moving two cams in rotation about a fixed axis;

b. moving one of said rotatable cams thereby effecting movement of said workpiece in an X direction;

c. moving the other one of said rotatable cams thereby effecting movement of said workpiece in a transverse direction;

1. said transverse movement of said workpiece being a pivoting-motion about a point to locate the instant center of curvature of said workpiece at all times on the feed line, said feed line is defined as the line through the grinding wheel center axis in the direction of compensation;

2. locating said point from said workpiece center axis by a distance which is derived by the form ula:

where the distance S is defined as the distance between the center axis of said workspindle (0) and the lower pivot shaft axis (61), 0 as the workpiece major radius, and 0-, as the workpiece minor radius.

i II 

1. An apparatus for grinding non-circular workpieces of revolution, comprising: a. a workhead having a holding means adapted to grip a workpiece with a major axis and a minor axis and rotate said workpiece about a center axis, said holding means mounted on a work spindle journalled in a work spindle housing (44); b. an X cam and a Y cam journalled for rotation on a shaft parallel to said work spindle, and rotating in timed relation therewith; c. a grinding wheel, said grinding wheel adapted to contact said workpiece at a point spaced from said center axis; d. a pair of X and Y cam followers attached to said work spindle housing and maintained in constant contact with said X and Y cams, respectively;
 1. said X cam follower adapted to move said workpiece in a direction toward or away from said grinding wheel, said direction defined as the X direction;
 2. said Y cam follower adapted to impart pivoting motion to said workhead about a point on said workhead, said point being located from said center axis of said workpiece a distance not less than the minor radius of said workpiece, nor greater than the major radius of said workpiece, in the direction of the contact point between said workpiece and said grinding wheel
 2. said Y cam follower adapted to impart pivoting motion to said workhead about a point on said workhead, said point being located from said center axis of said workpiece a distance not less than the minor radius of said workpiece, nor greater than the major radius of said workpiece, in the direction of the contact point between said workpiece and said grinding wheel
 2. A workhead for rotating about a center axis and shifting the axis of rotation of a non-circular workpiece having both a major and a minor axis, relative to a grinding wheel axis of rotation, comprising: a. a grinding wheel; b. a workspindle housing adapted to move in an X direction; c. said work spindle housing also adapted to be pivotally movable; d. X and Y cams rotated in timed relation to the rotation of said non-circular workpiece to produce work spindle housing movement; e. the axis of said pivotal motion being located from the center axis of said workpiece a distance not less than the workpiece minor radius nor greater than the workpiece major radius, in the direction of the contact point between said workpiece and said grinding wheel.
 2. Locating said point from said workpiece center axis by a distance which is derived by the formula: S Congruent (01 + 07)/2 where the distance S is defined as the distance between the center axis of said workspindle (0) and the lower pivot shaft axis (61), 01 as the workpiece major radius, and 07 as the workpiece minor radius.
 2. locating said point (61) by that distance from the center axis of said workpiece toward the contact point between said workpiece and said grinding wheel, which is equal to not less than the minor radius of said workpiece and not greater than the major radius of said workpiece.
 2. locating said point from said workpiece center axis by a distance which is derived by the formula: S Congruent (01 + 07)/2 where the distance S is defined as the distance between the center axis of said workspindle (0) and the lower pivot shaft axis (61), 01 as the workpiece major radius, and 07 as the workpiece minor radius.
 3. The invention defined in claim 1 wherein: a. the distance of said point from said workpiece center axis is structurally defined by the formula: S Congruent (01 + 07)/2 where the distance S is defined as the distance between the center axis of said workspindle (0) and the lower pivot shaft axis (61), 01 as the workpiece major radius, and 07 as the workpiece minor radius.
 4. The invention defined in claim 2, wherein: a. The distance of said centerline from said workpiece center axis is structurally defined by the formula: S Congruent (01 + 07)/2 where the distance S is defined as the distance between the center of axis of said workspindle (0) and the lower pivot shaft axis (61), 01 as the workpiece major radius, and 07 as the workpiece minor radius.
 5. A method of grinding a non-circular workpiece, having regions of reverse curvature with both a major axis and a minor axis, by rotating said workpiece about a center axis, comprising the steps of: a. moving two cams in rotation about a fixed axis; b. moving one of said rotatable cams thereby effecting movEment of said workpiece in an X direction; c. moving the other one of said rotatable cams thereby effecting movement of said workpiece in a transverse direction;
 6. A method of grinding a non-circular workpiece, having regions of reverse curvature with both a major axis and a minor axis, by rotating said workpiece about a center axis, comprising the steps of: a. moving two cams in rotation about a fixed axis; b. moving one of said rotatable cams thereby effecting movement of said workpiece in an X direction; c. moving the other one of said rotatable cams thereby effecting movement of said workpiece in a transverse direction;
 7. A method of grinding a non-circular workpiece having regions of reverse curvature, with both a major axis and a minor axis, by rotating said workpiece about a center axis, comprising the steps of: a. moving two cams in rotation about a fixed axis; b. moving one of said rotatable cams thereby effecting movement of said workpiece in an X direction; c. moving the other one of said rotatable cams thereby effecting movement of said workpiece in a transverse direction;
 8. The method defined in claim 7, wherein: a. locating said point from said workpiece center axis by a distance which is derived by the formula: S Congruent (01 + 07)/2 where the distance S is defined as the distance between the center axis of said workspindle (0) and the lower pivot shaft axis (61), 01 as the workpiece major radius, and 07 as the workpiece minor radius.
 9. A method of grinding a non-circular workpiece, having regions of reverse curvature with both a major axis and a minor axis, by rotating said workpiece about a center axis, comprising the steps of; a. moving two cams in rotation about a fixed axis; b. moving one of said rotatable cams thereby effecting movement of said workpiece in an X direction; c. moving the other one of said rotatable cams thereby effecting movement of said workpiece in a transverse direction; 